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Theorem rexnal2 3036
Description: Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
rexnal2 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)

Proof of Theorem rexnal2
StepHypRef Expression
1 rexnal 2989 . . 3 (∃𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 𝜑)
21rexbii 3034 . 2 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑)
3 rexnal 2989 . 2 (∃𝑥𝐴 ¬ ∀𝑦𝐵 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
42, 3bitri 264 1 (∃𝑥𝐴𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑥𝐴𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wral 2907  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-ral 2912  df-rex 2913
This theorem is referenced by:  rexnal3  3037  ralnex2  3038  isnsgrp  17209  tgdim01  25302  nn0prpw  31960  smprngopr  33483  clsk1independent  37826
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